Nonmetricity formulation of general relativity and its scalar-tensor extension

Einstein's celebrated theory of gravitation can be presented in three forms: general relativity, teleparallel gravity, and the rarely considered before symmetric teleparallel gravity. These formulations differ in the underlying geometric structure and interpretations, but are equivalent in observational predictions. The theories truly part when one extends them by e.g. nonminimally coupling a scalar field to the metric tensor degree of freedom. Thus extending symmetric teleparallel gravity, we introduce a new class of theories where a scalar field is coupled nonminimally to nonmetricity $Q$, which here encodes the gravitational effects like curvature $R$ in general relativity or torsion $T$ in teleparallel gravity. We derive the field equations and point out the similarities and differences with analogous scalar-curvature and scalar-torsion theories. We show that while scalar-nonmetricity gravity lacks invariance under conformal transformations, a suitable extra term can restore this; and also establish that $f(Q)$ gravity forms a particular subclass of scalar-nonmetricity theories. We illustrate the theory with an example of flat Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime.